Spatial Encryption: Improvements And Applications

In recent years, Identity-based encryption, or IBE for short, is a most pop-ular ?eld in cryptography. The setting greatly simpli?es the management ofcerti?cates in a traditional public-key system. In 2008, D. Boneh and M. Ham-burg developed a new framework, Generalized IBE, for constructing IBE systemwith di?erent properties. They also proposed a instance of GIBE called spatialencryption (SE). The scheme incorporates concept in linear algibra to providemore features. In such a system, a sender can encrypt a message under a vectorin Znp. Users hold secret key corresponding to a a?ine space. Decrypt works onlywhen the a?ine space contains the vector. In their paper, they show that manyimportant extensions for IBE system can be derived from spatial encryption.We will show in this paper how to improve the security of spatial encryption,and how to optimize the system in a pratical application. First, we give a briefreview of the history of IBE systems. We categorize them in their expressioncapability and construction technique. Then we present two alternative con-structions of spatial encryption. The ?rst one can be proved to be secure underthe decisional bilinear Di?ie-Hellman assumption, which is much simpler thanthe underlying BDHE assumption used in Boneh and Hamburg's construction.As a tradeo?, the size of a ciphertext in our scheme is linearly dependent on thespace dimension. We also discuss two techniques for reducing the private keysize. The second construction is a hybrid one. This scheme allows an e?iciencyparameterω(0≤ω≤1) during setup. A smallerωleads to e?iciency, while abigger one improves security. We notice that, by carefully investigating a realapplication, a hybrid scheme can achieve the optimal tradeo? between securityand e?iciency.